TY - JOUR

T1 - Analytical crossover results for the Feigenbaum constants: Crossover from conservative to dissipative systems

AU - Quispel, G. Reinout W.

PY - 1985

Y1 - 1985

N2 - Using renormalization theory, Zisook has shown that at small values of the dissipation (e.g., for a planar map of constant Jacobian B=0.999) the apparent rate of convergence, δ, of the parameter values for period doubling descends (‘‘crosses over’’) from its conservative value (δ=8.721 097 2. . .) to its dissipative value (δ=4.669 201 6. . .) as we consider higher and higher bifurcations. Moreover, this apparent δ at the nth bifurcation is a universal function, depending only on &. Employing a technique of Ghendrih and renormalization theory, I obtain an analytical expression for this function and show that it does not descend monotonically but has a minimum at Be,min≃3×10-6. I also analytically calculate the (orbit) scaling factor α at Be=0 and at Be=1 using some results of a second-order renormalization equation which I derive here. Finally, I obtain an analytical expression for the bifurcation values Cn(B) of a particular map (the Hénon map) as a function of n and B.

AB - Using renormalization theory, Zisook has shown that at small values of the dissipation (e.g., for a planar map of constant Jacobian B=0.999) the apparent rate of convergence, δ, of the parameter values for period doubling descends (‘‘crosses over’’) from its conservative value (δ=8.721 097 2. . .) to its dissipative value (δ=4.669 201 6. . .) as we consider higher and higher bifurcations. Moreover, this apparent δ at the nth bifurcation is a universal function, depending only on &. Employing a technique of Ghendrih and renormalization theory, I obtain an analytical expression for this function and show that it does not descend monotonically but has a minimum at Be,min≃3×10-6. I also analytically calculate the (orbit) scaling factor α at Be=0 and at Be=1 using some results of a second-order renormalization equation which I derive here. Finally, I obtain an analytical expression for the bifurcation values Cn(B) of a particular map (the Hénon map) as a function of n and B.

KW - IR-61287

U2 - 10.1103/PhysRevA.31.3924

DO - 10.1103/PhysRevA.31.3924

M3 - Article

VL - 31

SP - 3924

EP - 3928

JO - Physical review A : atomic, molecular, and optical physics and quantum information

JF - Physical review A : atomic, molecular, and optical physics and quantum information

SN - 2469-9926

IS - 6

ER -